Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458521 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16319 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
907 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,25,8,12,7,13)(3,9,26,31,16,33,17,32)(4,18,21,10,22,37,27,36)(5,11,23,41,24,39,28,40)(14,34,19,49,43,59,44,29)(15,30,48,57,46,58,47,60)(20,52,53,38,55,50,54,35)(42,62,51,56,45,61,63,64);(1,3,7,26,8,16,6,17)(2,9,25,32,12,33,13,31)(4,19,27,43,22,44,21,14)(5,15,28,48,24,46,23,47)(10,34,37,29,36,59,18,49)(11,30,41,60,39,58,40,57)(20,51,54,45,55,63,53,42)(35,61,52,56,38,62,50,64);(1,4,6,21,8,22,7,27)(2,10,13,18,12,36,25,37)(3,14,17,44,16,43,26,19)(5,20,23,53,24,55,28,54)(9,29,31,34,33,49,32,59)(11,38,40,52,39,35,41,50)(15,45,47,51,46,42,48,63)(30,56,57,61,58,64,60,62);(1,5)(2,11)(3,15)(4,20)(6,23)(7,28)(8,24)(9,30)(10,35)(12,39)(13,40)(14,42)(16,46)(17,47)(18,50)(19,51)(21,53)(22,55)(25,41)(26,48)(27,54)(29,56)(31,57)(32,60)(33,58)(34,61)(36,38)(37,52)(43,45)(44,63)(49,64)(59,62)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,c^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,d^c=de,g^c=gi,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 365718);